COMPOSITIO MATHEMATICA
Geometrically rational real conic bundles and
very transitive actions
J´er´emy Blanc and Fr´ed´eric Mangolte
Compositio Math. 147 (2011), 161–187.
doi:10.1112/S0010437X10004835
FOUNDATION COMPOSITIO MATHEMATICA
doi:10.1112/S0010437X10004835
Geometrically rational real conic bundles and
very transitive actions
J´er´emy Blanc and Fr´ed´eric Mangolte
Abstract
In this article we study the transitivity of the group of automorphisms of real algebraic surfaces. We characterize real algebraic surfaces with very transitive automorphism groups. We give applications to the classification of real algebraic models of compact surfaces: these applications yield new insight into the geometry of the real locus, proving several surprising facts on this geometry. This geometry can be thought of as a half-way point between the biregular and birational geometries.
1. Introduction
The group of automorphisms of a complex algebraic variety is small: indeed, it is finite in general. Moreover, the group of automorphisms is 3-transitive only if the variety is P1C. On the other hand, it was recently proved that for a surface X(R) birational to P2R, its group of automorphisms acts n-transitively on X(R) for any n. The main goal of this paper is to determine all real algebraic surfaces X(R) having a group of automorphisms which acts very transitively on X(R). For precise definitions and statements, see below.
The aim of this paper is to study the action of birational maps on the set of real points of a real algebraic variety. Let us emphasize a common terminological source of confusion about the meaning of what is a real algebraic variety (see also the enlightening introduction of [Kol01]). From the point of view of general algebraic geometry, a real variety X is a variety defined over the real numbers, and a morphism is understood as being defined over all the geometric points. In most real algebraic geometry texts, however, the algebraic structure considered corresponds to the algebraic structure of a neighbourhood of the real points X(R) in the whole complex variety; or, in other words, the structure of a germ of an algebraic variety defined over R.
From this point of view it is natural to view X(R) as a compact submanifold of Rn defined by real polynomial equations, where n is some natural integer. Likewise, it is natural to say that a map ψ : X(R) → Y (R) is an isomorphism if ψ is induced by a birational map Ψ : X 99K Y such that Ψ (respectively Ψ−1) is regular at any point of X(R) (respectively of Y (R)). In particular, ψ : X(R) → Y (R) is a diffeomorphism. This notion corresponds to the notion of biregular maps defined in [BCR98, 3.2.6] for the structure of real algebraic variety commonly used in the context of real algebraic geometry. To distinguish between the Zariski topology and the topology induced by the embedding of X(R) as a topological submanifold of Rn, we will call the latter the Euclidean topology. Throughout what follows, topological notions like connectedness or compactness will always refer to the Euclidean topology.
Received 19 June 2009, accepted in final form 2 March 2010, published online 13 September 2010.
Recall that a real projective surface is rational if it is birationally equivalent to the real projective plane, and that it is geometrically rational if its complexification is birationally equivalent to the complex projective plane. The number of connected components is a birational invariant. In particular, if X is a rational projective surface, then X(R) is connected.
The paper [HM09] proves that the group of automorphisms Aut(X(R)) acts n-transitively on X(R) for any n and any rational real algebraic surface X. To study the case where X(R) is not connected, we have to refine the notion of n-transitivity. Indeed, if X(R) has non-homeomorphic connected components, then even the group of self-homeomorphisms does not act 2-transitively.
Definition 0. Let G be a topological group acting continuously on a topological space M . We say that two n-tuples of distinct points (p1, . . . , pn) and (q1, . . . , qn) are compatible if there exists
an homeomorphism ψ : M → M such that ψ(pi) = qifor each i. The action of G on M is then said
to be very transitive if, for any pair of compatible n-tuples of points (p1, . . . , pn) and (q1, . . . , qn)
of M , there exists an element g ∈ G such that g(pi) = qi for each i. More generally, the action of
G is said to be very transitive on each connected component if we require the above condition only in the case when, for each i, pi and qi belong to the same connected component of M .
Up until now, it has not been known when the automorphism group of a real algebraic surface is big. We give a complete answer to this question: this is one of the main results of this paper. Let #M be the number of connected components of a compact manifold M .
Theorem 1. Let X be a non-singular real projective surface. The group Aut(X(R)) is then very transitive on each connected component if and only if X is geometrically rational and #X(R) 6 3.
In the three component case, Theorem2 below says that the very transitivity of Aut(X(R)) can be determined by examining the set of possible permutations of connected components. Theorem 2. Let X be a non-singular real projective surface. The group Aut(X(R)) then has a very transitive action on X(R) if and only if the following hold:
(i) X is geometrically rational; and (ii) (a) #X(R) 6 2; or
(b) #X(R) = 3, and there is no pair of homeomorphic connected components; or
(c) #X(R) = M1t M2t M3, M1∼ M26∼ M3, and there is a morphism π : X → P1R whose
general fibres are rational curves, and an automorphism of P1R which fixes π(M3) and
exchanges π(M1), π(M2); or
(d) #X(R) = M1t M2t M3, M1∼ M2∼ M3, and there is a morphism π : X → P1R whose
general fibres are rational curves, such that any permutation of the set of intervals {π(M1), π(M2), π(M3)} is realised by an automorphism of P1R.
Furthermore, when Aut(X(R)) is not very transitive, it is not even 2-transitive.
This theorem will be proved in §9. Note that, when #X(R) > 3, either any element of Aut(X(R)) preserves a conic bundle structure (Theorem 25), or Aut(X(R)) is countable (Corollary 11): in either case Aut(X(R)) is not 1-transitive.
These two theorems apply to the classification of algebraic models of real surfaces. Up to this point in the paper X(R) is considered as a submanifold of some Rn. Conversely, let M be a compact C∞-manifold. By the Nash–Tognoli theorem [Tog73], every such M is diffeomorphic to a non-singular real algebraic subset of Rm for some m. Taking the Zariski closure in Pm and
applying Hironaka’s resolution of singularities [Hir64], it follows that M is in fact diffeomorphic to the set of real points X(R) of a non-singular projective algebraic variety X defined over R. Such a variety X is called an algebraic model of M . A natural question is to classify the algebraic models of M up to isomorphism for a given manifold M .
There are several recent results about algebraic models and their automorphism groups [BH07, HM09, HM10, KM09]. For example, when M is two-dimensional, and admits a real rational algebraic model, this rational algebraic model is unique [BH07]. In other words, if X and Y are two rational real algebraic surfaces, then X(R) and Y (R) are isomorphic if and only if they are homeomorphic. We extend the classification of real algebraic models to geometrically rational surfaces.
Theorem 3. Let X, Y be two non-singular geometrically rational real projective surfaces, and assume that #X(R) 6 2. The surface X(R) is then isomorphic to Y (R) if and only if X is birational to Y and X(R) is homeomorphic to Y (R). This is false in general when #X(R) > 3.
Recall that a singular projective surface is minimal if any birational morphism to a non-singular surface is an isomorphism. We have the following rigidity result on minimal geometrically rational real surfaces.
Theorem 4. Let X and Y be two minimal geometrically rational real projective surfaces, and assume that either X or Y is non-rational. The following are then equivalent.
(i) X and Y are birational.
(ii) X(R) and Y (R) are isomorphic.
In this work, we classify the birational classes of real conic bundles and correct an error contained in the literature (Theorem25). It follows that the only geometrically rational surfaces X(R) for which equivalence by homeomorphism implies equivalence by isomorphism are the connected ones. In particular, this yields a converse to [BH07, Corollary 8.1].
Corollary 5. Let M be a compact C∞-surface. The surface M then admits a unique geometrically rational model if and only if the following two conditions hold:
(i) M is connected; and
(ii) M is non-orientable or M is orientable with genus g(M ) 6 1.
For M orientable with g(M ) > 1, no uniqueness result, even a very weak one, holds. We can therefore ask what the simplest algebraic model for such an M should be. This question will be studied in a forthcoming paper by J. Huisman and F. Mangolte.
Another way of measuring the size of Aut(X(R)) was used in [KM09], where it was proved that, for any rational surface X, Aut(X(R)) ⊂ Diff(X(R)) is dense for the strong topology. For non-geometrically rational surfaces and for most of the non-rational geometrically rational surfaces the group Aut(X(R)) cannot be dense. The above paper left the question of density open only for certain geometrically rational surfaces with two, three, four or five connected components. One by-product of our results is the non-density of Aut(X(R)) for most surfaces with at least three connected components; see Proposition 41.
Let us mention some other papers on automorphisms of real projective surfaces. In [RV05], it is proved that Aut(P2(R)) is generated by linear automorphisms and certain real algebraic automorphisms of degree five. The paper [HM10] is devoted to the study of very transitive actions and uniqueness of models for some kinds of singular rational surfaces.
1.1 Strategy of the proof
In the proof of Theorem 1, the main part concerns minimal conic bundles. We first prove that two minimal conic bundles are isomorphic if they induce the same intervals on the basis. Given a set of intervals, one can choose the most special conic bundle, the so-called exceptional conic bundle, to write explicitly the automorphisms and to obtain a fibrewise transitivity. We then use the most general conic bundles which come with distinct foliations on the same surface. The foliations being transversal, this yields the very transivity of the automorphism group in the minimal case.
1.2 Outline of the article
In §2 we fix the notation and in §3we recall the classification of minimal geometrically rational real surfaces.
Section 4, which constitutes the technical heart of the paper, is devoted to conic bundles, especially minimal ones. We give representative elements of isomorphism classes, and explain the links between the various conic bundles.
In §5, we investigate real surfaces which admit two conic bundle structures. In particular, we show that these are del Pezzo surfaces, and give descriptions of the possible conic bundles on these surfaces. Section6is devoted to the proof of Theorem4. We firstly correct an inaccuracy in the literature, by proving that if two surfaces admitting a conic bundle structure are birational, then the birational map may be chosen so that it preserves the conic bundle structures. We then strengthen this result to isomorphisms between real parts when the surfaces are minimal, before proving Theorem 4.
In §7, we prove that if the real part of a minimal geometrically rational surface has two or three connected components, then its automorphism group is very transitive on each connected component. In §8, we prove the same result for non-minimal surfaces. We show how to separate infinitely close points, which is certainly one of the most counter-intuitive aspects of our geometry, and was first observed in [BH07] for rational surfaces. We also prove the uniqueness of models in many cases.
In §9, we then use all the results of the preceding sections to prove the main results stated in the introduction (except Theorem 4, which is proved in §6).
2. Notation
Throughout what follows, by a variety we will mean an algebraic variety, which may be real or complex (i.e. defined over R or C). If the converse is not expressly stated all our varieties will be projective and all our surfaces will be non-singular and geometrically rational (i.e. rational over C).
Recall that a real variety X may be identified with a pair (S, σ), where S is a complex variety and σ is an anti-holomorphic involution on S; by abuse of notation we will write X = (S, σ). Then, S(C) = X(C) denotes the set of complex points of the variety, and X(R) = S(C)σ is the set of real points. A point p ∈ X may be real (if it belongs to X(R)), or imaginary (if it belongs to X(C)\X(R)). If X(R) is non-empty (which will be the case for all our surfaces), then Pic(X) ∼= Pic(S)σ (see [Sil89, I.(4.5)]). As we only work with regular surfaces (i.e. q(X) = q(S) = 0), the Picard group is isomorphic to the N´eron–Severi group, and ρ(S) and ρ(X) will denote the rank of Pic(S) and Pic(X), respectively. Recall that ρ(X) 6 ρ(S).
We denote by KX ∈ Pic(X) the canonical class, which may be identified with KS. The
intersection of two divisors of Pic(S) or Pic(X) will always denote the usual intersection in Pic(S). We will use the classical notions of morphisms, rational maps, isomorphisms and automorphisms between real or complex varieties. Moreover, if X1 and X2 are two real varieties,
an isomorphism between real parts X1(R) ψ
→ X2(R) is a birational map ψ : X199K X2such that ψ (respectively ψ−1) is regular at any point of X1(R) (respectively of X2(R)). This endows X1(R)
with a structure of a germ of algebraic variety defined over R (as in [BCR98, 3.2.6]), whereas the structure of X1 is that of an algebraic variety.
This notion of isomorphism between real parts gives rise to a geometry with rather unexpected properties compared to those of the biregular geometry or the birational geometry. For example, let α : X1(R) → X2(R) be an isomorphism, and ε : Y199K X1, η : Y299K X2 be two birational
maps; the map ψ := ε−1αη is a well-defined birational map. Then ψ can be an isomorphism Y1(R) → Y2(R) even if neither ε nor η is an isomorphism between real parts. In the same vein, let
α : X1(R) → X2(R) be an isomorphism, and let η1: Y1→ X1 and η2: Y2→ X2 be two birational
morphisms which are the blow-ups of only real points (which may be proper or infinitely near points of X1 and X2). If α sends the points blown-up by η1 to the points blown-up by η2, then
β = (η2)−1αη1: Y1(R) → Y2(R) is an isomorphism.
Using Aut and Bir to denote respectively the group of automorphisms and birational self-maps of a variety, we have the following inclusions for the groups associated to X = (S, σ):
Aut(S) ⊂ Bir(S)
∪ ∪
Aut(X) ⊂ Aut(X(R)) ⊂ Bir(X).
By Pn we mean the projective n-space, which may be complex or real depending on the context. It is unique as a complex variety, written as PnC. However, as a real variety, Pn may either be PnCendowed with the standard anti-holomorphic involution, written PnR, or, only when n is odd, PnCwith a special involution with no real points, written as (Pn, ∅). To lighten notation, and since we never speak about (P1, ∅)(R), we write P1(R) for P1R(R).
3. Minimal surfaces and minimal conic bundles
The aim of this section is to reduce our study of geometrically rational surfaces to surfaces which admit a minimal conic bundle structure. We first recall the classification of geometrically rational surfaces (see [Sil89] for an introduction). The proofs of Theorems2and4 will then split into three cases: rational, del Pezzo with ρ = 1, and minimal conic bundle. The rational case is treated in [HM09] and Proposition 10 below states the case of a del Pezzo surface with ρ = 1. Definition 6. A conic bundle is a pair (X, π) where X is a surface and π is a morphism X → P1, where any fibre of π is isomorphic to a plane conic. If (X, π) and (X0, π0) are two conic bundles, a birational map of conic bundles ψ : (X, π) 99K (X0, π0) is a birational map ψ : X 99K X0 such that there exists an automorphism α of P1 with π0◦ φ = π ◦ α.
We will assume throughout what follows that if X is real, then the basis is P1R(and not (P1, ∅)). This avoids certain conic bundles with no real points. We denote by Aut(X, π) (respectively Bir(X, π)) the group of automorphisms (respectively birational self-maps) of the conic bundle (X, π). Observe that Aut(X, π) = Aut(X) ∩ Bir(X, π). Similarly, when (X, π) is real we denote by Aut(X(R), π) the group Aut(X(R)) ∩ Bir(X, π).
Recall that a real algebraic surface X is minimal if and only if there is no real (−1)-curve and no pair of disjoint conjugate imaginary (−1)-curves on X, and that a real conic bundle (X, π) is minimal if and only if the two irreducible components of any real singular fibre of π are imaginary. Compare this to the complex case where (X, π) is minimal if and only if there is no singular fibre.
The following two results follow from the work of Comessatti [Com12] (see also [Isk79,Man67], [Sil89, ch. V], or [Kol]). Recall that a surface X is a del Pezzo surface if the anti-canonical divisor −KX is ample. The same definition applies for X real or complex.
Theorem 7. If X is a minimal geometrically rational real surface such that X(R) 6= ∅, then one and exactly one of the following holds:
(i) X is rational: it is isomorphic to P2R, to the quadric Q0:= {(x : y : z : t) ∈ P3R| x2+ y2+ z2=
t2}, or to a real Hirzebruch surface Fn, n 6= 1;
(ii) X is a del Pezzo surface of degree one or two with ρ(X) = 1;
(iii) there exists a minimal conic bundle structure π : X → P1 with an even number of singular fibres 2r > 4; moreover, ρ(X) = 2.
Remark 8. If (S, σ) is a minimal geometrically rational real surface such that S(C)σ= ∅, then S is an Hirzebruch surface of even index.
Proposition 9 (Topology of the real part). In each case of the former theorem, we have: (i) X is rational if and only if X(R) is connected. When X is moreover minimal, then X(R) is
homeomorphic to one of the following: the real projective plane, the sphere, the torus, or the Klein bottle;
(ii) when X is a minimal del Pezzo surface of degree one, it satisfies ρ(X) = 1, and X(R) is the disjoint union of one real projective plane and four spheres. If X is a minimal del Pezzo surface of degree two with ρ(X) = 1, then X(R) is the disjoint union of four spheres; (iii) if X is non-rational and is endowed with a minimal conic bundle with 2r singular fibres,
then X(R) is the disjoint union of r spheres, r > 2.
Proposition 10. Let X, Y be two minimal geometrically rational real surfaces. Assume that X is not rational and satisfies ρ(X) = 1 (but ρ(Y ) may be equal to 1 or 2).
(i) If X is a del Pezzo surface of degree one, then any birational map X 99K Y is an isomorphism. In particular,
Aut(X) = Aut(X(R)) = Bir(X).
(ii) If X is a del Pezzo surface of degree two, X is birational to Y if and only if X is isomorphic to Y . Moreover, all the base-points of the elements of Bir(X) are real, and
Aut(X) = Aut(X(R)) ( Bir(X).
Proof. Assume the existence of a birational map ψ : X 99K Y . If ψ is not an isomorphism, we decompose ψ into elementary links
X = X0 ψ1 99K X1 ψ2 99K · · ·ψ99K Xn−1 n−1 ψn 99K Xn= Y
as in [Isk96, Theorem 2.5]. It follows from the description of the links of [Isk96, Theorem 2.6] that, for any link ψi: Xi−199K Xi, Xi−1and Xi are isomorphic del Pezzo surfaces of degree two,
is a del Pezzo surface of degree one, α ∈ Aut(X0) is the Bertini involution of the surface, and β : Xi+1→ Xi is an isomorphism.
Therefore, Y is isomorphic to X. Moreover, if X has degree one, then ψ is an isomorphism. If X has degree two, then ψ is decomposed into conjugates of Bertini involutions, so each of its base-points is real. This proves that if ψ ∈ Aut(X(R)), then ψ ∈ Aut(X). Furthermore, conjugates of Bertini involutions belong to Bir(X) but not to Aut(X) = Aut(X(R)). 2 Corollary 11. Let X0 be a minimal non-rational geometrically rational real surface with
ρ(X0) = 1, and let η : X → X0 be a birational morphism.
Then, Aut(X(R)) is countable. Moreover, if X0 is a del Pezzo surface of degree one, then
Aut(X(R)) is finite.
Proof. Without changing the isomorphism class of X(R) we may assume that η is the blow-up of only real points (which may belong to X0 as proper or infinitely near points). Since any
base-point of any element of Bir(X0) is real (Proposition 10), the same is true for any element
of Bir(X). In particular, Aut(X(R)) = Aut(X). The group Aut(X) acts on Pic(X) ∼= Zn, where n = ρ(X) > 1. This action gives rise to an homomorphism θ : Aut(X) → GL(n, Z). Let us prove that θ is injective. Indeed, if α ∈ Ker(θ), then α is conjugate by η to an element of α0∈ Aut(X0)
which acts trivially on Pic(X0). Writing S0 as the complex surface obtaining by forgetting the
real structure of X0, S0 is the blow-up of seven or eight points in the general position of P2C.
Thus α0∈ Aut(X0) ⊂ Aut(S0) is the lift of an automorphism of P2C which fixes seven or eight
points, with no three collinear, and hence is the identity.
The morphism θ is injective, and this shows that Aut(X(R)) = Aut(X) is countable. Moreover, if X0is a del Pezzo surface of degree one, then Bir(X0) = Aut(X0) (by Proposition10).
Since Aut(X0) is finite, Aut(X(R)) ⊂ Bir(X) is also finite. 2
4. Minimal and exceptional conic bundles
Definition 12. If (X, π) is a real conic bundle, I(X, π) ⊂ P1(R) denotes the image by π of the set X(R) of real points of X.
The set I(X, π) is the union of a finite number of intervals (which may be ∅ or P1(R)), and it is well-known that it determines the birational class of (X, π). We prove that I(X, π) also determines the equivalence class of (X(R), π) among the minimal conic bundles, and give the proof of Theorem4 in the case of conic bundles (Corollary20).
Lemma and Definition 13. Let (X, π) be a real minimal conic bundle. The following conditions are equivalent.
(i) There exists a section s such that s and ¯s do not intersect.
(ii) There exists a section s such that s2= −r, where 2r is the number of singular fibres. If any of these conditions occur, we say that (X, π) is exceptional.
Proof. Let s be a section satisfying one of the two conditions. Denote by (S, π) the complex conic bundle obtained by forgetting the real structure of (X, π), and by η : X → Fm the birational
map which contracts in any singular fibre of π the irreducible component which does not intersect s. If s satisfies condition(i), η(¯s) and η(s) are two sections of Fmwhich do not intersect,
so they have self-intersections −m and m. This means that s2= ¯s2= −m and that the number of singular fibres is 2m, and implies(ii). Conversely, if s satisfies(ii), η(s) and η(¯s) are sections of Fm of self-intersection −r and r. If these two sections are distinct, they do not intersect, which
means that s and ¯s do not intersect. If η(s) = η(¯s), we have r = 0, and X = (P1C× P1
C, σ) for a
certain anti-holomorphic involution σ. We may thus choose another section s0 of self-intersection
0 which is imaginary. 2
Remark 14. The definition of exceptional conic bundles was introduced in [Bla09b, DI09] for complex conic bundles endowed with an holomorphic involution. If (S, π) is an exceptional complex conic bundle with at least four singular fibres, then Aut(S, π) = Aut(S) is a maximal algebraic subgroup of Bir(S) [Bla09b].
Lemma 15. Let (Y, πY) be a minimal real conic bundle such that πY has at least one singular
fibre. There exists an exceptional real conic bundle (X, πX) and an isomorphism ψ : Y (R) →
X(R) such that πX ◦ ψ = πY.
Remark 16. The result is false without the assumption on the number of singular fibres. Consider for example Y = F3(R), whose real part is homeomorphic to the Klein bottle. Indeed, any
exceptional conic bundle with no singular fibres is a real form of (P1C× P1
C, pr1), and thus has a
real part either empty or homeomorphic to the torus S1× S1.
Before proving Lemma 15, we associate to any given exceptional conic bundle X an explicit circle bundle isomorphic to it. The following improves [Sil89, Corollary VI.3.1] where the model is only assumed to be birational to X.
Lemma 17. Let (X, π) be an exceptional real conic bundle. Then, there exists an affine real variety A ⊂ X isomorphic to the affine surface of R3 given by
y2+ z2= Q(x),
where Q is a real polynomial with only simple roots, all real. Moreover, π|A: A → P1R is the
projection (x, y, z) 7→ (x : 1), and I(X, π) is the closure of {(x : 1) ∈ P1R | Q(x) > 0}.
Furthermore, if f = π−1((1 : 0)) ⊂ X is a non-singular fibre, then the singular fibres of π are those of the points {(x : 1) | Q(x) = 0} and the inclusion A → X is an iso-morphism A(R) → (X\f ) (R). In particular, if (1 : 0) /∈ I(X, π), then the inclusion yields an isomorphism A(R) → X(R).
Proof. Denote by 2r the number of singular fibres of π (which is even, see Lemma 13). Assume first that r = 0, which implies that (X, π) is a real form of (P1C× P1
C, pr1), and hence is isomorphic to (P1R× P1 R, pr1) or to (P 1 R× (P 1, ∅), pr
1); see the convention after Definition 6.
Taking Q(x) = 1 or Q(x) = −1 gives the result.
Assume now that r > 0, and denote by s and ¯s two conjugate imaginary sections of π of self-intersection −r. Changing π by an automorphism of P1, we can assume that (1 : 0). The singular fibres of π are above the points (a1: 1), . . . , (a2r: 1), where the ai are distinct real numbers.
Let J = (J1, J2) be a partition of {a1, . . . , a2r} into two sets of r points. Let η be the birational
morphism (not defined over R) which contracts the irreducible component of π−1((ai: 1)) which
intersects s if ai∈ J1 and the component which intersects ¯s if ai∈ J2. Then, the images of s and
¯
s are two sections of self-intersection 0. Thus we may assume that η is a birational morphism of conic bundles (S, π) → (P1C× P1
the real structure of X and pr1 is the projection on the first factor, and where η(s) and η(¯s) are equal to P1
C× (0 : 1) and P 1
C× (1 : 0).
We write P1(x1, x2) =Qa∈J1(x1− ax2) and P2(x1, x2) =
Q
a∈J2(x1− ax2), and denote by α
and σ the self-maps of S, which are the lifts by η of the following self-maps of P1 C× P
1 C:
α0: ((x1: x2), (y1: y2)) 99K ((x1: x2), (−y2· P1(x1, x2) : y1· P2(x1, x2))),
σ0: ((x1: x2), (y1: y2)) 99K ((x1: x2), (−y2· P1(x1, x2) : y1· P2(x1, x2))).
The map α0 is a birational involution of P1C× P1
C, which is defined over R, and whose
base-points are precisely the 2r base-points {((x : 1), (0 : 1)) | x ∈ J1} ∪ {((x : 1), (1 : 0)) | x ∈ J2} blown-up
by η. Since α0 is an involution and η is the blow-up of all of its base-points, α = η−1α0η is an automorphism of S, which belongs to Aut(S, π). In consequence, σ is an anti-holomorphic involution of S.
Denote by σX the anti-holomorphic involution on S which gives the real structure of X.
The map σX ◦ σ−1 belongs to Aut(S, π) and acts trivially on the basis, since σ and σX
have the same action on the basis. Moreover, since both σX and σ exchange the irreducible
components of each singular fibre, σX ◦ σ−1 preserves any curve contracted by η and is therefore
the lift by η of β : ((x1: x2), (y1: y2)) 7→ ((x1: x2), (µy1: y2)) for some µ ∈ C∗. It follows that
σX0 = η ◦ σX◦ η−1= β ◦ σ0 is the map
σX0 : ((x1: x2), (y1: y2)) 99K ((x1: x2), (−µ·y2P1(x1, x2) : y1P2(x1, x2))).
Let us write Q(x) = −µP1(x, 1)P2(x, 1), and denote by B ⊂ C3 the affine hypersurface of
equation y2+ z2= Q(x), and by πB: B → P1 the map (x, y, z) 7→ (x : 1). Let A = (B, σB), where
σB sends (x, y, z) onto (¯x, ¯y, ¯z). Denote by θ : B 99K P1C× P1C the map that sends (x, y, z) onto
((x : 1), (y − iz : P2(x, 1))) if P2(x, 1) 6= 0 and onto ((x : 1), (−µP1(x, 1) : y + iz)) if P1(x, 1) 6= 0.
Then θ is a birational morphism, and θ−1 sends ((x1: x2), (y1: y2)) on
x1 x2 ,1 2 y1 y2 P2(x1, x2) − y2 y1 µP1(x1, x2) , i 2 y1 y2 P2(x1, x2) + y2 y1 µP1(x1, x2) . Observe that σ0Xθ = σBθ. In consequence, ψ = η−1◦ θ is a real birational map A 99K X.
Moreover, ψ is an isomorphism from B to the complement in S of the union of π−1((1 : 0)) and the pull-back by η of P1× (0 : 1) and P1× (1 : 0). Indeed let x
0∈ C. If x0∈ C is such
that Q(x0) 6= 0, then θ restricts to an isomorphism from πB−1((x0: 1)) to {((x0: 1), (y1: y2)) ∈
P1C× P1C | y1y26= 0} ∼= C∗. If Q(x0) = 0, then x0∈ J1∪ J2, and the fibre πB−1((x0: 1)) consists of
two lines of C2 which intersect, given by y = iz and y = −iz. If x0∈ J1, then the line y + iz = 0
is sent isomorphically by θ onto the fibre {((x0: 1), (y1: y2)) ∈ P1C× P1C | y26= 0} ∼= C∗, and the
line y − iz is contracted on the point ((x0: 1), (0 : 1)). The map ψ thus sends isomorphically
πB−1((x0: 1)) onto the fibre π−1((x0: 1)) minus the two points corresponding to the two sections
whose self-intersection is −r. The situation when x0∈ J2 is similar.
The map ψ is therefore an inclusion A → X and, by construction, it satisfies all the properties
stated in the lemma. 2
Proof of Lemma 15. Take a section s of πY. If s intersects its conjugate ¯s at a real point p
(respectively into a pair of imaginary points q1 and q2), then blow-up the point p (respectively
q1 and q2), and contract the strict transform of the fibre of the blown-up point(s). Repeating
this process, we obtain a minimal real conic bundle (Z, πZ) and a birational map φ : Y 99K Z
If all the base-points of φ are imaginary, we set ψ = φ and (X, πX) = (Z, πZ). Otherwise, by
induction on the number of real base-points of φ, it suffices to prove the existence of ψ when φ is an elementary link centred at only one real point.
Denote by q ∈ Z the real point which is the base-point of φ−1. Since πY has at least one
singular fibre, this is also the case for πZ, and thus I(Z, πZ) is not the whole P1(R) (by
Lemma 17). We may thus assume that (1 : 0) /∈ I(Z, πZ), that πZ(q) = (1 : 1), and that the
interval of I(Z, πZ) which contains πZ(q) is {(x : 1) ∈ P1R | 0 6 x 6 a} for some a ∈ R, a > 1.
Take the affine surface A ⊂ Z given by Lemma 17, which is isomorphic to y2+ z2= Q(x) for some polynomial Q. Then, Q(0) = Q(a) = 0 and Q(x) > 0 for 0 < x < a, and we may assume that Q(1) = 1. Denote by s the section of πZ: Z → P1Rgiven locally by y + iz = ixn, for some positive
integer n. Its conjugate is given by y − iz = −ixn, or y + iz = Q(x)/(−ixn). Thus, s intersects ¯s at some real point p ∈ Z, its image x = πZ(p) satisfies Q(x)/(−ixn) = ixn, or Q(x) = x2n. Taking
n large enough, this can only happen when x = 0 or x = 1. The first possibility cannot occur since a section does not pass through the singular point of a singular fibre. Thus, s intersects ¯s at only one real point, which is q. In consequence, the strict pull-back by φ of s is a section of Y which intersects its conjugate at only imaginary points. This shows that (Y (R), πY) is isomorphic to
an exceptional real conic bundle (X, πX). 2
Corollary 18. Let (X, πX) and (Y, πY) be two minimal real conic bundles, and assume that
either πX or πY has at least one singular fibre. Then, the following are equivalent:
(i) I(X, πX) = I(Y, πY);
(ii) there exists an isomorphism ϕ : X(R) → Y (R) such that πY ◦ ϕ = πX.
Proof. It suffices to prove(i)⇒(ii). By Lemma15, we may assume that both (X, πX) and (Y, πY)
are exceptional. We may now assume that the fibre over (1 : 0) is not singular and use Lemma17: let AX ⊂ X and BX ⊂ Y be the affine surfaces given by the lemma, with equations y2+ z2=
QX(x) and y2+ z2= QY(x) respectively. Since I(X, πX) = I(Y, πY), QY(x) = λQX(x) for some
positive λ ∈ R. The map (x, y, z) 7→ (x,√λy,√λz) then yields an isomorphism (X(R), πX) →
(Y (R), πY). 2
The above result implies the next two corollaries. The first one strengthens a result of Comessatti [Com12] (see also [Kol, Theorem 4.5]).
Corollary 19. Let (X, π) and (X0, π0) be two real conic bundles. Assume that (X, π) and (X, π0) are minimal. Then (X(R), π) and (X0(R), π0) are isomorphic if and only if there exists an automorphism of P1R that sends I(X, π) onto I(X0, π0). 2 Corollary 20. Let (X, πX) and (Y, πY) be two minimal conic bundles. Then, the following
are equivalent:
(i) (X(R), πX) and (Y (R), πY) are isomorphic;
(ii) (X, πX) is birational to (Y, πY) and X(R) is isomorphic to Y (R).
Proof. The implication (i)⇒ (ii)is evident. Let us prove the converse.
Since (X, πX) is birational to (Y, πY) and both of them are minimal, the number of singular
fibres of πX and πY is the same, being equal to 2r for some non-negative integer r.
Assume that r = 0, which means that X is an Hirzebruch surfaces Fm for some m and that
prove that (X(R), π) and (Y (R), π) are isomorphic, by taking elementary links at two imaginary distinct fibres (see for example [Man06, Proof of Theorem 6.1]).
When r > 0, already the fact that (X, πX) is birational to (Y, πY) implies that (X(R), πX)
is isomorphic to (Y (R), πY) (Corollary 18). 2
5. Conic bundles on del Pezzo surfaces
In this section, we focus on surfaces admitting distinct minimal conic bundles. We will see that these surfaces are necessarily del Pezzo surfaces (Lemma23). We begin by the description of all possible minimal real conic bundles occurring on del Pezzo surfaces.
Lemma 21. Let V be a subset of P1(R); then the following are equivalent:
(i) there exists a minimal real conic bundle (X, π) with I(X, π) = V such that X is a del Pezzo surface;
(ii) the set V is a union of closed intervals, and #V 6 3.
Proof. The implication(i) ⇒ (ii) is easy. Indeed, if (X, π) is minimal, it is well-known that the number of singular fibres of π is even, denoted 2r, and that 2r = 8 − (KX)2. Since −KX is ample,
KX2 > 1, and thus r 6 3. The conclusion follows as I(X, π) is the union of r closed intervals. Let us prove the converse. If V = P1(R) or V = ∅, we take (X, π) to be (P1C× P1
C, pr1), where
pr1 is the projection on the first factor, endowed with the anti-holomorphic map that sends
((x1: x2), (y1: y2)) onto ((x1: x2), (±y2: y1)).
Now we can assume that V consists of k closed intervals I1, . . . , Ik, with 1 6 k 6 3. For
j = 1, . . . , 3, we denote by mj an homogenous form of degree two. If j 6 k, we choose that mj
vanishes at the boundary of the interval Ij, and is non-negative on Ij. If j > k, we choose mj
such that mj is positive on P1(R). In any case, we choose that m1· m2· m3has six distinct roots.
We consider the real surface given by X := {((x : y : z), (a : b)) ∈ P2R× P1
R | x 2m
1(a, b) + y2m2(a, b) + z2m3(a, b) = 0}.
The projection on P2Ris a double covering. A straightforward calculation shows that this covering is ramified over a smooth quartic. In consequence, X is a smooth surface, and precisely a del Pezzo surface of degree two. Taking π : X → P1R as the second projection, we obtain a conic bundle (X, π) on the del Pezzo surface X such that I(X, π) = V . If k = 3, the conic bundle is minimal. Otherwise, we contract components in the imaginary singular fibres (corresponding to the roots of mj for j > k) to obtain the result. 2
Recall the following classical result, which will be useful throughout what follows.
Lemma 22. Let π : S → P1Cbe a complex conic bundle, and assume that S is a del Pezzo surface, with (KS)2= 9 − m 6 7. Then, there exists a birational morphism η : S → P2Cwhich is a blow-up
of m points p1, . . . , pm and which sends the fibres of π onto the lines passing through p1. The
curves of self-intersection −1 of S are:
– the exceptional curves η−1(p1), . . . , η−1(pm);
– the strict transforms of the lines passing through two of the pi;
– the conics passing through five of the pi;
Proof. Denote by ε the contraction of one component in each singular fibre of π. Then, ε is a birational morphism of conic bundles, not defined over R, from S to a del Pezzo surface which is also an Hirzebruch surface. Changing the contracted components, we may assume that ε is a map S → F1. Contracting the exceptional section onto a point p1∈ P2C, we get a birational map
η : S → P2
C which is the blow-up of m points p1, . . . , pm of P 2
C, and which sends the fibres of π1
onto the lines passing through p1. The description of the (−1)-curves is well-known and may be
found for example in [Dem80]. 2
Lemma 23. Let π1: X → P1Rbe a minimal real conic bundle. Then, the following conditions are
equivalent.
(i) There exist a real conic bundle π2: X → P1R, such that π1 and π2 induce distinct foliations
on X(C).
(ii) Either X is isomorphic to P1R× P1
R, or X is a del Pezzo surface of degree two or four.
Moreover, if the conditions are satisfied, then the following occur. (a) The map π2 is unique, up to an automorphism of P1R.
(b) There exist α ∈ Aut(X) and β ∈ Aut(P1R) such that π1α = βπ2. Moreover, if X is a del Pezzo
surface of degree two, then α may be chosen to be the Geiser involution.
(c) Denoting by f1, f2⊂ Pic(X) the divisors of the general fibre of respectively π1 and π2, we
have f1+ f2= −cKX where c = 4/(KX)2∈ N ·12.
Proof. We now prove that condition(i)implies condition(ii), (a) and (c). Assuming the existence of π2, we denote by fi the divisor of the fibre of πi for i = 1, 2. We have (f1)2= (f2)2= 0 and by
the adjunction formula f1· KX = f2· KX = −2, where KX is the canonical divisor. Let us write
d = (KX)2.
Since (X, π1) is minimal, Pic(X) has rank two, and hence f1= aKX+ bf2, for some a, b ∈ Q.
Computing (f1)2 and f1· KX we find respectively 0 = a2d − 4ab = a(ad − 4b) and −2 = ad − 2b.
If a = 0, we find f1= f2, a contradiction. Thus, 4b = ad and 2b = ad + 2, which yields b = −1
and ad = −4, so f1+ f2= −4/d · KX. This shows that f2 is uniquely determined by f1, which is
the assertion (a).
Denote as usual by S the complex surface associated to X. Let C ∈ Pic(S) be an effective divisor, with reduced support, and let us prove that C · (f1+ f2) > 0. Since C is effective,
C · f1> 0 and C · f2> 0. If C · f1= 0, then the support of C is contained in one fibre of π1.
If C is a multiple of f1, then C · f2> 0; otherwise, C is a multiple of a (−1)-curve contained in a
singular fibre of f1, and the orbit of C by the anti-holomorphic involution is equal to a multiple
of f1, whence C · f2> 0.
Since f1+ f2 is ample, and f1+ f2= −4/d · KX, either KX or −KX is ample. The surface
X being geometrically rational, the former cannot occur, whence d > 0. If S is isomorphic to P1
C× P 1
C, the existence of π1, π2 shows that X is isomorphic to P 1 R× P
1 R.
Otherwise, KX is not a multiple in Pic(XC) and thus d is equal to 1, 2 or 4. The number of
singular fibres being even and equal to 8 − (KX)2, the only possibilities are then 2 and 4.
We have proved that condition(i) implies condition(ii), (a) and (c). Assume now that X = (S, σ) is P1R× P1
R or a del Pezzo surface of degree two or four. We
construct an automorphism α of X which does not belong to Aut(X, π). Then, by taking π2= π1α, we get assertion(i). Taking into account the unicity of π2, we get assertion (b).
If X is P1R× P1
R, the two conic bundles are given by the projections on each factor, and we
can get for α the swap of the factors.
If X is a del Pezzo surface of degree two, the anti-canonical map ζ : X → P2 is a double covering ramified along a smooth quartic, cf. for example [Dem80]. Let α be the involution associated to the double covering; α is classically called the Geiser involution. It fixes a smooth quartic, and hence cannot preserve any conic bundle.
The remaining case is when X is a del Pezzo surface of degree four. By Lemma 22, there is a birational map η : S → P2C which is the blow-up of five points p1, . . . , p5 of P2C, no three
being collinear, and which sends the fibres of π1 on the lines passing through p1 . There are 16
exceptional curves (curves isomorphic to P1C of self-intersection (−1)) on S: – E1= η−1(p1), . . . , E5= η−1(p5) (five curves);
– the strict transforms of the lines passing through pi and pj, denoted by Lij (10 curves);
– the strict transform of the conic passing through the five points, denoted by Γ.
Note that the four singular fibres of π1 are Ei∪ Lij, i = 2, . . . , 5, and that σ exchanges thus
Ei and Lij for i = 1, . . . , 5. The intersection form being preserved, this implies that σ acts on
the 16 exceptional curves as
(E2L12)(E3L13)(E4L14)(E5L15)(E1Γ)(L23L45)(L24L35)(L25L34).
After a linear change of coordinates, we may assume that p1= (1 : 1 : 1), p2= (1 : 0 : 0),
p3= (0 : 1 : 0), p4= (0 : 0 : 1) and p5= (a : b : c) for some a, b, c ∈ C∗. Denote by φ the birational
involution (x : y : z) 99K (ayz : bxz : cxy) of P2C. Since the base-points of φ are p2, p3, p4 and since
φ exchanges p1 and p5, the map α = η−1φη is an automorphism of S. Its action on the 16
exceptional curves is given by the permutation
(L23E4)(L24E3)(L34E2)(L12L25)(L13L35)(L14L45)(ΓL15)(E1E5).
Observe that the actions of α and σ on the set of 16 exceptional curves commute. This means that ασα−1σ−1 is an holomorphic automorphism of S which preserves any of the 16 curves. It is the lift of an automorphism of P2Cthat fixes the five points p1, . . . , p5 and hence is the identity.
Consequently, α and σ commute, so α ∈ Aut(X). Since φ sends a general line passing though p1
onto a conic passing through p2, . . . , p5, α belongs to Aut(X)\Aut(X, π). 2
Corollary 24. Let X be a minimal geometrically rational real surface, which is not rational. Then, the following are equivalent.
(i) #X(R) = 2 or #X(R) = 3.
(ii) There exists a geometrically rational real surface Y (R) isomorphic to X(R), and such that Y admits two minimal conic bundles π1: Y → P1Rand π2: Y → P1Rinducing distinct foliations
on Y (C).
Proof. (ii) ⇒ (i). By Lemma 23, Y is then a del Pezzo surface, which has degree two or four since Y is not rational. This implies that #Y (R) = 2 or #Y (R) = 3 by Proposition 9.
(i)⇒(ii). According to Theorem7, Proposition9implies the existence of a minimal real conic bundle structure πX : X → P1Rwith four or six singular fibres. This condition is equivalent to the
fact that I(X, πX) is the union of two or three intervals. According to Lemma 21, there exists
Corollary19 shows that (X(R), πX) and (Y, π1) are isomorphic. Moreover, Lemma23yields the
existence of π2. 2
6. Equivalence of surfaces versus equivalence of conic bundles
This section is devoted to the proof of Theorem 4. From Theorem 7 and Proposition 10, it remains to solve the conic bundle case, which is done in Theorem 27. First of all, we correct an existing inaccuracy in the literature; in [Kol, Exercice 5.8] or [Sil89, VI.3.5], it is asserted that all minimal real conic bundles with four singular fibres belong to a unique birational equivalence class. To the contrary, the following general result, which includes the case with four singular fibres, occurs.
Theorem 25. Let πX : X → P1R and πY : Y → P1R be two real conic bundles, and suppose that
either X or Y is non-rational. Then, the following are equivalent. (i) The two real surfaces X and Y are birational.
(ii) The two real conic bundles (X, πX) and (Y, πY) are birational.
(iii) There exists an automorphism of P1 which sends I(X, πX) onto I(Y, πY).
Moreover, if the number of singular fibres of πX is at least eight, then Bir(X) = Bir(X, πX).
Remark 26. It is well-known that this result is false when X and Y are rational. Indeed, consider (X, πX) = (P1R× P1R, pr1) and let (Y, πY) be a real conic bundle with two singular fibres. The
surfaces X and Y are birational, but the conic bundles (X, πX) and (Y, πY) are not.
Proof. The equivalence(iii) ⇔ (ii)was proved in Corollary19 and (ii)⇒ (i) is evident.
Let us now prove the equivalence (i) ⇔ (ii). We may assume that (X, πX) and (Y, πY) are
minimal and that X is not rational; hence πX has at least four singular fibres. Let ψ : X 99K Y
be a birational map, and decompose ψ into elementary links: ψ = ψn◦ · · · ◦ ψ1 (see [Isk96,
Theorem 2.5]). Consider the first link ψ1: X 99K X1, which may be of type (II) or (IV) only,
by [Isk96, Theorem 2.6]. If ψ1 is of type (II), then ψ1 is a birational map of conic bundles
(X, πX) 99K (X1, π1) for some conic bundle structure π1: X1→ P1. If ψ1 is of type (IV), then
ψ1 is an isomorphism X → X1 and the link is precisely a change of conic bundle structure from
πX to π1: X1→ P1, which induces distinct foliations on X(R). Applying Lemma 23, X is a del
Pezzo surface of degree two or four, and there exist automorphisms α ∈ Aut(X) and β ∈ Aut(P1R) such that π1ψ1α = βπ2, whence (X, π) is isomorphic to (X1, π1). We proceed by induction on
the number of elementary links to conclude that (X, πX) is birational to (Y, πY). Moreover, if
πX has at least eight singular fibres, then no link of type (IV) may occur, so ψ is a birational
map of conic bundles (X, πX) 99K (Y, πY). 2
When the conic bundles are minimal, we can strengthen Theorem 25to get an isomorphism between the real parts.
Theorem 27. Let πX : X → P1R and πY : Y → P1R be two minimal real conic bundles, and
suppose that either X or Y is non-rational. Then, the following are equivalent. (i) X and Y are birational.
(ii) X(R) and Y (R) are isomorphic.
Proof. The implications(iii)⇒ (ii)⇒ (i)being evident, it suffices to prove (i)⇒ (iii). Since X and Y are not rational, both πX and πY have at least one singular fibre. Applying Lemma 15,
we may assume that both (X, πX) and (Y, πY) are exceptional real conic bundles. Then, since
(X, πX) and (Y, πY) are birational (Theorem25), we may assume that I(X, πX) = I(Y, πY), up
to an automorphism of P1
R. Then Corollary19 shows that (X, πX) is isomorphic to (Y, πY). 2
We are now able to prove Theorem 4concerning minimal surfaces.
Proof of Theorem 4. Let X and Y be two minimal geometrically rational real surfaces, and assume that either X or Y is non-rational.
If X(R) and Y (R) are isomorphic, it is clear that X and Y are birational. Let us prove the converse.
Theorem7lists all the possibilities for X. If ρ(X) = 1 or ρ(Y ) = 1, Proposition10shows that X is isomorphic to Y . Otherwise, since neither X nor Y is rational, there exist minimal conic bundle structures on X and on Y . From Theorem27, we conclude that X(R) is isomorphic to
Y (R). 2
To go further with non-minimal surfaces, we need to know when the group Aut(X(R)) is very transitive for X minimal. This is done in the following sections.
7. Very transitive actions
Thanks to the work done in §4, it is easy to apply the techniques of [HM09] to prove that Aut(X(R)) is fibrewise very transitive on a real conic bundle. After describing the transitivity of Aut(X(R)) on the tangent space of a general point, we set the main result of this section: Aut(X(R)) is very transitive on each connected component when X is minimal and admits two conic bundle structures (Proposition 33). We end the section by giving a characterisation of surfaces X for which Aut(X(R)) is able to mix the connected components of X(R).
Lemma 28. Let (X, π) be a minimal real conic bundle over P1R with at least one singular fibre. Let (p1, . . . , pn) and (q1, . . . , qn) be two n-tuples of distinct points of X(R), and let (b1, . . . , bm)
be m points of I(X, π). Assume that π(pi) = π(qi) for each i, that π(pi) 6= π(pj) for i 6= j and
that π(pi) 6= bj for any i and any j.
Then, there exists α ∈ Aut(X(R)) such that α(pi) = qi for every i, πα = π and α|π−1(b
i)is the
identity for every i.
Remark 29. The same result holds for minimal real conic bundles with no singular fibre, see [BH07, 5.4]. The following proof uses twisting maps, see below, which were introduced in [HM09] to prove that the action of the group of automorphisms Aut(S2) on the quadric sphere S2:= {(x : y : z) ∈ R3 | x2+ y2+ z2= 1} is very transitive.
Proof. By Lemma15, we may assume that (X, π) is exceptional. Moreover, Lemma17yields the existence of an affine real surface A ⊂ X isomorphic to the hypersurface of R3 given by
y2+ z2= −
2r
Y
i=1
(x − ai),
for some a1, . . . , a2r∈ R with a1< a2< · · · < a2r, where π|A corresponds to the projection
For i = 1, . . . , n, let us denote by (xi, yi, zi) the coordinates of piin A ⊂ R3 and by (ui, vi, wi)
those of qi. From the hypothesis, we have xi= ui for all i; thus we get yi2+ zi2= vi2+ w2i for all i.
Let Φi∈ SO2(R) be the rotation sending (xi, yi) to (ui, vi). Then, by [HM09, Lemma 2.2], there
exists an algebraic map Φ : [a1, a2r] → SO2(R) such that Φ(xi) = Φi for i = 1, . . . , n and Φ(bi)
is the identity for i = 1, . . . , m. Let us recall the proof; since SO2(R) is isomorphic to the unit
circle S1:= {(x : y : z) ∈ P2(R) | x2+ y2= z2}, it suffices to prove the statement for S1 instead of
SO2(R). Let Φ0 be a point of S1 distinct from Φ1, . . . , Φnand from the identity. Since S1\{Φ0}
is isomorphic to R, it suffices, finally, to prove the statement for R instead of SO2(R). The latter
statement is an easy consequence of Lagrange polynomial interpolation.
Then the map defined by α : (x, y, z) 7→ (x, (y, z) · Φ(x)) induces an automorphism A(R) → A(R) called the twisting map of π associated to Φ. Moreover, α(pi) = qi, for all i, πα = π, α|π−1(b
i)
is the identity for every i, and π induces an automorphism X(R) → X(R). 2 Lemma 30. Let (X, π) be a minimal real conic bundle over P1Rwith at least one singular fibre. Let p ∈ X be a real point in a non-singular fibre of π, and let Σ ⊂ I(X, π) be a finite subset, with π(p) ∈ Σ. Denote by η : Y → X the blow-up of p, and by E ⊂ Y the exceptional curve. Let q ∈ E be the point corresponding to the direction of the fibre of π passing through p.
Then, the lift of the group
G = {α ∈ Aut(X(R)), πα = π | α|π−1(Σ) is the identity}
by η is a subgroup η−1Gη ⊂ Aut(Y (R)) which fixes the point q, and acts transitively on E\q ∼= A1R.
Proof. Since G acts identically on π−1(Σ), it fixes p, and therefore lifts to H = η−1Gη ⊂ Aut(Y (R), πη), which preserves E. Moreover, G preserves the fibre of π passing through p, so H preserves its strict transform, which intersects transversally E at q, so q is fixed.
Let us prove now that the action of η−1Gη on E\q is transitive. By Lemma 15, we may assume that (X, π) is exceptional. Then, we take an affine surface A ⊂ X, isomorphic to the hypersurface y2+ z2= P (x) of R3 for some polynomial P , such that A|
π is the projection
prx: (x, y, z) 7→ x and the inclusion A ⊂ X gives an isomorphism A(R) → X(R) (Lemma 17). Let us write (x0, y0, z0) ∈ R3 as the coordinates of p. Since x is on a non-singular fibre of π, then
P (x0) > 0. Up to an affine automorphism of R3, and up to multiplication of P by some constant,
we may assume that x0= 0, P (0) = 1, y0= 0, and z0= 0.
To any real polynomial λ ∈ R[X], we associate the matrix
α(X) β(X) −β(X) α(X)
∈ SO2(R(X)),
where α = (1 − λ2)/(1 + λ2) ∈ R(X) and β = 2λ/(1 + λ2) ∈ R(X). Moreover, corresponding to this matrix, we associate the map
ψλ: (x, y, z) 7→ (x, α(x) · y − β(x) · z, β(x) · y + α(x) · z),
which belongs to Aut(A(R), prx). To impose that ψλ is the identity on (prx)−1(Σ) is the same
as asking that λ(x) = 0 for each (x : 1) ∈ Σ ⊂ P1(R), and in particular for x = 0.
Denote by O = R[x, y, z]/(y2+ z2− P (x)) the ring of functions of A, by p ⊂ O the ideal of functions vanishing at p, by Op the localisation, and by m ⊂ Op the maximal ideal of Op. Then,
of x, y, z − 1 ∈ R[x, y, z]. Since P (0) = 1, we may write P (x) = 1 + xQ(x), for some real poly-nomial Q. We compute
[0] = [y2+ z2− P (x)] = [y2+ (z − 1)2+ 2(z − 1) − xQ(x)] = [2(z − 1) − xQ(0)] ∈ m/m2. We see that [z − 1] = [xQ(0)/2], and thus m/m2 is generated by [x] and [y] as an R-module. Since λ(0) = 0, we can write λ(x) = xµ(x), for some real polynomial µ. The linear action of ψλ
on the cotangent space Tp,A∗ fixes [x] and sends [y] onto [α(x) · y − β(x) · z] = (1 − λ(x) 2)y − 2λ(x)z λ(x)2+ 1 = [y − 2λ(x)(1 + xQ(0)/2)] = [y − 2µ(0)x].
It suffices to change the derivative of λ at 0 (which is equal to µ(0)), which may be any real number. Therefore, the action of G on the projectivisation of Tp,A∗ fixes a point (corresponding to [x]) but acts transitively on the complement of this point. Since E corresponds to the projectivisation of Tp,A, G acts transitively on E\q. 2
Lemma 31. Let X be a real projective surface endowed with two minimal conic bundles π1: X → P1Rand π2: X → P1Rinducing distinct foliations on X(C). There exists a real projective
surface X0 such that X0(R) and X(R) are isomorphic, X is endowed with two minimal conic bundles π01: X → P1R and π02: X → P1R inducing distinct foliations on X0(C) and the following condition holds.
(?) Let Fj be a real fibre of πj0, j = 1, 2. If F1(R) ∩ F2(R) 6= ∅, then at most one of the curves
Fj can be singular.
Remark 32. It is possible that the condition (?) above does not hold for (X, π1, π2), taking, for
example, for X the del Pezzo surface of degree two given in the proof of Lemma21 for k = 3: X := {((x : y : z), (a : b)) ∈ P2R× P
1 R | x
2m
1(a, b) + y2m2(a, b) + z2m3(a, b) = 0}.
The map π1: X → P1Ris given by the second projection, and the 6 singular points of its singular
fibres correspond to only three points of P2R, namely (1 : 0 : 0), (0 : 1 : 0) and (0 : 0 : 1). This shows that the Geiser involution preserves the set of the six points, so each of these points is the singular point of a singular fibre of π2.
Proof. Suppose that the condition (?) does not hold for (X, π1, π2) (otherwise, the result is
obvious). Then Fi is the union of two (−1)-curves Ei,1 and Ei,2, intersecting transversally at
some point pi. Since pi is the only real point of Fi, we have p1= p2. Hence, F1· E2,i> 2 for
i = 1, 2, which implies that F1· F2> 4. According to Lemma 23, X is a del Pezzo surface of
degree two or four, and we have F1+ F2= −cKX with c = 4/(KX)2. Computing 16/(KX)2=
(F1+ F2)2= 2F1· F2> 8, we see that (KX)2= 2.
Let q ∈ X(R) be a real point, let η : Y → X be the blow-up of q, and let ε : Y → X0 be the contraction of the strict transform of the fibre of π1 passing through q. Let ψ : X 99K X0 be
the composition ψ = ε ◦ η−1. We prove now that if q is general enough, then X0 is a del Pezzo surface of degree two, and π01= π1◦ ψ and π20 = σX0◦ π01 (where σX0 ∈ Aut(X0) is the Geiser
involution of X0) satisfy the condition (?).
Firstly, it is well-known that blowing-up a general point of a del Pezzo surface of degree two yields a del Pezzo surface of degree one (it suffices that q does not belong to any of the
(−1)-curves of X and to the ramification curve of the double covering X → P2); then a contraction from a del Pezzo surface of degree one yields a del Pezzo surface of degree two.
Secondly, we denote respectively by S, S0, T the complex surfaces obtained by forgetting the real structures of X, X0, Y and study condition (?) by working now in the Picard groups of these surfaces, identifying a curve with its equivalence class. We choose a (−1)-curve (not defined over R) in any of the six singular fibres of π10, and denote these by C1, . . . , C6, and
denote by p1, . . . , p6 the singular points of the six singular fibres, so that pi∈ Ci. Condition
(?) amounts to prove that Di:= ψ−1σX0ψ(Ci) ⊂ S does not pass through pj for any i and
any j. Fixing i and j, we will see that this yields a curve of X where q should not lie. Note that the action of the Geiser involution σX0∈ Aut(X0) ⊂ Aut(S0) on Pic(S0) is given by
σX0(D) = (D · KX0)KX0 − D (this follows directly from the fact that the invariant part of Pic(S0)
has rank one). In consequence, the (−1)-curve D0i:= σX0ψ(Ci) ⊂ S0 is equal to −KX0 − ψ(Ci),
and thus ε∗(Di) = −ε∗(KX0) − η∗(Ci). Writing Eq as the (−1)-curve contracted by η, and f
as a general fibre of π1, the (−1)-curve contracted by ε is equivalent to η∗(f ) − Eq. We have
KY = η∗(KX) + Eq= ε∗(KX0) + η∗(f ) − Eq in Pic(Y ). This implies that
η∗(Di) = ε∗(Di0) = −η∗(KX) + η∗(f ) − η∗(Ci) − 2Eq∈ Pic(Y ).
This means that Di is a curve with a double point at q, is equivalent to −KX + f − Ci∈
Pic(S) and has self-intersection 3. Moreover, the linear system Λi of curves in Pic(S) equivalent
to −KX + f − Cihas dimension three. Note that Λidoes not depend on q, but only on i. Denote
by Λi,j⊂ Λi the sublinear system of curves of Λi passing through pj. This system has dimension
two; after blowing-up pj, the system Λi,j yields a ramified double covering of P2. If Di passes
through pj, then Dicorresponds to a member of Λi,j, singular at q, and this implies that q belongs
to the ramified locus of the double covering induced by Λi,j. It suffices to choose q outside of all
these loci to obtain condition (?). 2
We now use the above lemmas to show that the action of Aut(X(R)) is very transitive on each connected component when X is a surface with two conic bundles.
Proposition 33. Let X be a real projective surface, which admits two minimal conic bundles π1: X → P1R and π2: X → P1R inducing distinct foliations on X(C).
Let (p1, . . . , pn) and (q1, . . . , qn) be two n-tuples of distinct points of X(R) such that pi
and qi belong to the same connected component for each i. Then, there exists an element of
Aut(X(R)) which sends pi onto qi for each i, and which sends each connected component
of X(R) onto itself.
Proof. When X is rational, the result follows from [HM09, Theorem 1.4]. Thus we assume that X is non-rational, and in particular that X(R) is non-connected.
From Lemma 31, we can assume that any real point which is critical for one fibration is not critical for the second fibration. Otherwise (recall that the fibrations are minimal), a real intersection point of a fibre F1 of π1 with a fibre F2 of π2 cannot be a singular point of F1
and of F2 at the same time. By Lemma 28 applied to (X, π1), and to (X, π2), we may assume
without loss of generality that all points p1, . . . , pn, q1, . . . , qn belong to smooth fibres of π1
and to smooth fibres of π2. We now use Lemma28 to obtain an automorphism α of (X(R), π1)
such that π2(α(pi)) 6= π2(α(pj)) and π2(α(qi)) 6= π2(α(qj)) for i 6= j. Hence, we may suppose that
Likewise, using an automorphism of (X(R), π2) we may suppose that π1(pi) 6= π1(pj) and
π1(qi) 6= π1(qj) for i 6= j.
We now show that for i = 1, . . . , m there exists an element αi∈ Aut(X(R)) that sends pi on
qi and that restricts to the identity on the sets
S
j6=i{pj} and
S
j6=i{qj}. Then, the composition
of the αi will achieve the proof. Observe that ζ = π1× π2 gives a finite surjective morphism
X → P1R× P1
Rwhich is two-to-one or four-to-one depending on the degree of X (this follows from
assertion (c) of Lemma23). Denote by W the image of X(R). The map X(R) → W is a differential map, which has topological finite degree. Denote by Wi the connected component of W which
contains both ζ(pi) and ζ(qi). Observe that Wi is contained in the square I(X, π1) × I(X, π2),
and that, for each point x ∈ Wi, the intersection of the horizontal and vertical lines (fibres of
the two projections of P1R× P1
R) passing through x with Wi is either only {x}, when x is on the
boundary of Wi, or is a bounded interval. Moreover, Wi is connected. Then, there exists a path
from ζ(pi) to ζ(qi) which is a sequence of vertical or horizontal segments contained in Wi. We may
furthermore assume that none of the segments is contained in (pr1)−1(π1(a)) or (pr2)−1(π2(a))
for any a ∈ (S
j6=i{pj}) ∪ (Sj6=i{qj}). Denote by r1, . . . , rl the points of U that are sent on the
singular points or ending points of the path, and by s1, . . . , sl some points of X(R) which are
sent by ζ on r1, . . . , rl respectively. Up to renumbering, s1= pi, sl= qi and two consecutive
points sj and sj+1 are such that π1(sj) = π1(sj+1) or π2(sj) = π2(sj+1). We then construct αi as
a composition of l − 1 maps, each one belonging either to Aut(X(R), π1) or Aut(X(R), π2) and
sending sj on sj+1, and fixing the points (Sj6=i{pj}) ∪ (Sj6=i{qj}). 2
The following proposition describes the possible mixes of connected components.
Proposition 34. Let (X, π) be a minimal real conic bundle. Denote by I1, . . . , Ir the r
connected components of I(X, π), and by M1, . . . , Mr the r connected components of X(R),
where Ii= π(Mi), Mi= π−1(Ii) ∩ X(R). If ν ∈ Symris a permutation of {1, . . . , r}, the following
are equivalent:
(i) there exists α ∈ Aut(P1R) such that α(Ii) = Iν(i) for each i;
(ii) there exists β ∈ Aut(X(R), π) such that β(Mi) = Mν(i) for each i;
(iii) there exists β ∈ Aut(X(R)) such that β(Mi) = Mν(i) for each i;
(iv) there exist two real Zariski open sets V, W ⊂ X, and β ∈ Bir(X), inducing an isomorphism V → W , such that β(V (R) ∩ Mi) = W (R) ∩ Mν(i) for each i.
Moreover, the conditions are always satisfied when r 6 2, and are in general not satisfied when r > 3.
Proof. The implications(ii) ⇒(i) and (ii)⇒ (iii) ⇒ (iv)are obvious. The implication(i) ⇒ (ii)is a direct consequence of Corollary 19.
We now prove that if r 6 2, then assertion(i)is always satisfied, and hence all the conditions are equivalent (since all are true). When r 6 1, take α to be the identity. When r = 2, we make a linear change of coordinates to the effect that I1= {(x : 1) | 0 6 x 6 1} and I2 is bounded by
(1 : 0) and (λ : 1), for some λ ∈ R, λ > 1 or λ < 0. Then, α : (x1: x2) 7→ (λx2: x1) is an involution
which exchanges I1 and I2.
It remains to prove the implication (iv) ⇒ (i) for r > 3. We decompose β into elementary links X = X0 β1 99K X1 β2 99K · · ·β99K Xn−1 n−1 βn 99K Xn= X
as in [Isk96, Theorem 2.5]. It follows from the description of the links of [Isk96, Theorem 2.6] that each of the links is of type (II) or (IV), and that the links of type (II) are birational maps of conic bundles and the links of type (IV) occur on del Pezzo surfaces of degree two.
In consequence, each of the Xi admits a conic bundle structure given by πi: Xi→ P1R,
where π0= πn= π, and if βi has type (II), then it is a birational map of conic bundles
(Xi−1, πi−1) 99K (Xi, πi), and if it has type (IV), then it is an isomorphism Xi−1→ Xi which
does not send the general fibre of πi−1on those of πi. In this latter case, since πiand πi−1βi have
distinct general fibres, Xi−1and Xiare del Pezzo surfaces of degree two, and the Geiser involution
ιi−1∈ Aut(Xi−1) exchanges the two general fibres (this follows from [Isk96, Theorem 2.6], but
also from Lemma23). This means that the map βi◦ ιi−1, that we denote by γi, is an isomorphism
of conic bundles (Xi−1, πi−1) → (Xi, πi).
Now, we prove by induction on the number of links of type (IV) that β may be decomposed into compositions of elements of Bir(X, π) and maps of the form ψιψ−1 where ψ is a birational map of conic bundles (X, π) 99K (X0, π0), (X0, π0) is a del Pezzo surface of degree two and ι ∈ Aut(X0) is the Geiser involution. If there is no link of type (IV), β preserves the conic bundle structure given by π. Otherwise, denote by βi the first link of type (IV), which is an
isomorphism βi: Xi→ Xi+1, and write βi= γi◦ ιi−1 as before. We write ψ = βi−1◦ · · · ◦ β1,
which is a birational map of conic bundles ψ : (X, π) 99K (Xi, πi). Then, β = (βn◦ · · · ◦ βi+1◦ γi◦
ψ)(ψ−1ιi−1ψ). Applying the induction hypothesis on the map (βn◦ · · · ◦ βi+1◦ γi◦ ψ) ∈ Bir(X),
we are done.
Now, observe that when (X0, π0) is a minimal real conic bundle and X0 is a del Pezzo surface of degree two, the map ζ : X0→ P2
Rgiven by |−KX0| is a double covering, ramified over a smooth
quartic curve Γ ⊂ P2R(see e.g. [Dem80]). Since (X, π) is minimal, (KX)2= 8 − 2r and thus π has
r = 6 singular fibres , so I(X, π) is the union of three intervals and X(R) is the union of three connected components. This implies that Γ(R) is the union of three disjoint ovals. A connected component M of X(R) is homeomorphic to a sphere, and surjects by ζ to the interior of one of the three ovals. The Geiser involution (induced by the double covering) induces an involution on M , which fixes the preimage of the oval. This means that the Geiser involution sends any connected component of X(R) on itself. Thus, in the decomposition of β into elements of Bir(X, π) and conjugate elements of Geiser involutions, the only relevant elements are those of Bir(X, π). There thus exists β0∈ Bir(X, π) which acts on the connected components of X(R) in the same way as β. This shows that assertion (iv)implies (i).
We finish by proving that assertion (i) is false in general, when r > 3. This follows from the fact that if Σ is a general finite subset of 2r distinct points of P1R, then the group {α ∈ Aut(P1
R) | α(Σ) = Σ} is trivial. Supposing this fact to be true, we obtain the result by
applying it to the 2r boundary points of I(X, π). Let us prove the fact. The set of 2r-tuples of P1R is an open subset W of (P1R)2r. For any non-trivial permutation υ ∈ Sym2r, we denote by Wυ⊂ W
the set of points a = (a1, . . . , a2r) ∈ W such that there exists α ∈ Aut(P1R) with α(ai) = aυ(i) for
each i. Let a ∈ Wυ, and take two 4-tuples Σ1, Σ2 of ai with Σ16= Σ2 and Σ2= υ(Σ1) (this is
possible since υ is non-trivial). Then, the cross-ratios of the ai in Σ1 and in Σ2 are the same.
This implies a non-trivial condition on W . Consequently, Wυ is contained in a closed subset
of W . Doing this for all non-trivial permutations υ, we obtain the result. 2
8. Real algebraic models
The aim of this section is to go further with non-minimal surfaces with two or three connected components. We begin by showing how to separate infinitely near points to the effect that
any such a surface Y (R) is isomorphic to a blow-up Ba1,...,amX(R) where X is minimal and
a1, . . . , amare distinct proper points of X(R). Then, we replace X(R) by an isomorphic del Pezzo
model (Corollary 24) and we use the fact that Aut(X(R)) is very transitive on each connected component for such an X (Proposition33) to prove that in many cases, if two birational surfaces Y and Z have homeomorphic real parts, then Y (R) and Z(R) are isomorphic. As a corollary, we get that in any case Aut(Y (R)) is very transitive on each connected component.
Proposition 35. Let X be a minimal geometrically rational real surface, with #X(R) = 2 or #X(R) = 3, and let η : Y → X be a birational morphism.
Then there exists a blow-up η0: Y0→ X, whose centre is a finite number of distinct real proper points of X, and such that Y0(R) is isomorphic to Y (R).
Moreover, we can assume that the isomorphism Y (R) → Y0(R) induces an homeomorphism η−1(M ) → (η0)−1(M ) for each connected component M of X(R).
Proof. According to Corollary 24, we may assume that X admits two minimal conic bundles π1: X → P1R and π2: X → P1R inducing distinct foliations on X(C). Preserving the isomorphism
class of Y (R), we may assume that the points in the centre of η are all real (such a point may be a proper point of X(R) or an infinitely near point). Let us denote by m (= KX2 − KY2) the
number of those points. We prove the result by induction on m.
The cases m = 0 and m = 1 being obvious (take η0= η), we assume that m > 2. We decompose η as η = θ ◦ ε, where ε : Y → Z is the up of one real point q ∈ Z, and θ : Z → Y is the blow-up of m − 1 real points. By the induction hypothesis, we may assume that θ is the blow-blow-up of m − 1 proper points of X, namely a1, . . . , am−1∈ X(R). Moreover, applying Proposition 33,
we may move the points by an element of Aut(X(R)), and assume that π1(ai) 6= π1(aj) and
π2(ai) 6= π2(aj) for i 6= j, and that the fibre of π1 passing through ai and the fibre of π2 passing
through ai are non-singular and transverse at ai, for each i.
If θ(q) /∈ {a1, . . . , am−1}, then η is the blow-up of m distinct proper points of X, and hence we
are done. Otherwise, assume that θ(q) = a1. We write E = θ−1(a1) ⊂ Z, and denote by Fi⊂ Z
the strict pull-back by η of the fibre of πi passing through a1, for i = 1, 2. Then, F1 and F2
are two (−1)-curves which do not intersect. Hence, the point q ∈ E belongs to at most one of the two curves, so we may assume that q /∈ F1. Denote by θ2: Z → X2 the contraction of the
m − 1 disjoint (−1)-curves F1, θ−1(a2), . . . , θ−1(am−1). Since q does not belong to any of these
curves, η2= θ2◦ ε is the blow-up of m − 1 distinct proper points of X2. It remains to find an
isomorphism γ : X2(R) → X(R) such that, for each connected component M of X(R), γη2 sends
η−1(M ) on M .
Denoting π0= π1◦ θ ◦ θ−12 , the map ψ = θ2◦ θ−1 is a birational map of conic bundles
(X, π1) 99K (X2, π0), which factorizes as the blow-up of a1, followed by the contraction of the strict
transform of the fibre passing through a1. Therefore, the conic bundle (X2, π0) is minimal. Since
#X(R) > 1 and π0ψ = π1, Corollary18yields the existence of an isomorphism γ : X2(R) → X(R)
such that π1γ = π0. Observe that γη2◦ η−1= γθ2◦ θ−1= γψ is a birational map X 99K X which
satisfies π ◦ (γη2◦ η−1) = π. Consequently, for any connected component M of X(R),
which corresponds to π−1(V ) ∩ X(R), for some interval V ⊂ P1R, we find π(γη2η−1(M )) =
π(M ) = V ; thus γη2 sends η−1(M ) on M . 2
Corollary 36. Let X be a minimal geometrically rational real surface, such that #X(R) = 2 or #X(R) = 3, and let η : Y → X, ε : Z → X be two birational morphisms. Denote by M1, . . . , Mr
